Questions tagged [matching]

A matching (aka **Independent Edge Set**) in a simple graph is the set of pairwise non-adjacent edges i.e. no two edges have common vertex.

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1answer
180 views

Game on the graph with matchings

The game on the graph $G$ is defined as follows. Initially, the chip is located at one of the vertices (let's call it the starting one). The players take turns, on each move it is necessary to move ...
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11 views

Maximal vs maximum matchings

Let $M_1$ be an inclusion-maximal matching in $G$ (that is, there is no matching which strictly contains it), and $M_2$ a maximum-size matching in $G$. How to prove that $|M_2| \le 2|M_1|$?
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38 views

Independent sets into which all the vertices of the graph can be split

How to prove that if $G$ is an acyclic transitive digraph, then the least independent sets into which all vertices of G can be divided is equal to the size of the longest paths to $G$?
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23 views

Set of cycles in directed graph

I have a directed graph. How to find in it some set of cycles that are pairwise do not intersect, but cover the entire set of vertices, if a cycle from one vertex is not considered a cycle, but cycle ...
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1answer
21 views

Maximum one-to-many matching

Let $G = (X+Y,E)$ be a bipartite graph and $k\geq 1$ an integer. A maximum $k$-matching is a subset of $E$ in which each vertex of $X$ is adjacent to at most $k$ edges and each vertex of $Y$ is ...
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44 views

Combinatorial Problem similar in nature to a special version of max weighted matching problem

I have a problem and want to know if there is any combinatorial optimization that is similar in nature to this problem or how to solve this special version of the max weight matching problem. I have a ...
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45 views

Why did Hopcroft and Karp write $M_0, M_1, M_2, \cdots, M_i, \cdots$? (Hopcroft - Karp Algorithm)

I am reading “An $n^{\frac{5}{2}}$ Algorithm for Maximum Matchings in Bipartite Graphs” by Hopcroft and Karp. Please see the image below. Let $s$ be the cardinality of a maximum matching. I think any ...
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1answer
180 views

Weighted Online Matching - randomized algorithms

Let's consider the edge weighted online matching problem. The Vertices arrive online and reveal all their current edges and edge-weights $w_e>0$. The goal is to maximize the matchings weight. An ...
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1answer
35 views

Is there such a problem as b-Matching with different b values?

Consider a bipartite Garph $G=(L \cup R, E)$. Naturally, a b-Matching problem is to find a set of edges $M \subset E$, such that each node in $L$ and $R$ are adjuscent to maximum $b$ neighbors, and a ...
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15 views

How to generate a uniform random sample of unique vertex pairings from a undirected graph under constraint?

I'm working on a research project where I have to pair up entities together and analyze outcomes. Normally, without constraints on how the entities can be paired, I could easily select one random ...
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A request for literature on Matching your partner problem

I need reference for a good book (a title and an author will do) or reference on the web which explains the problem of matching procedures of suitable partners between several males and females based ...
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1answer
80 views

Are these problems NP-Complete?

I got 2 decision problem that I need to answer if they are in P or they are NP complete: 1.Just like subset sum: given the integers or natural numbers ${\displaystyle w_{1},\ldots ,w_{n}}$ does ...
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23 views

Stable matching with dynamic preference lists

I have a set $F$ of $n_1$ families, a set $C$ of $n_2$ children ($n_1<n_2$) and a set $M$ of feasible one-to-one matchings of the families with the children. All the children have the same ...
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20 views

Blossom Algorithm

I have noticed that in the blossom algorithm, we need to find the blossom while finding the augmenting path. What will happen if I just know there is a blossom in a graph and the contracted graph has ...
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46 views

Special case of stable marriage

I have an instance of the stable marriage problem in which the first side $S_1$ has $n_1$ agents and the second side $S_2$ has $n_2$ agents with $n_2$ is very big in comparison to $n_1$. In addition, ...
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30 views

Options for approaching stable marriage problem with unequally sized sets of elements/preferences

I am looking for an algorithm/code that will provide stable matching for two unequally sized sets of elements (clubs and students) with an unequal set of preferences. There is a large pool of students ...
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1answer
76 views

Selecting k rows and k columns from a matrix to maximize the sum of the k^2 elements

Suppose $A$ is an $n \times n$ matrix, and $k \ge 1$ is an integer. We want to find $k$ distinct indices from $\{1, 2, \ldots, n\}$, denoted as $i_1, \ldots, i_k$, such that $\sum_{p, q = 1}^k A_{...
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Algorithm for b-matching on bipartite graph [duplicate]

I have a bipartite graph, where I want to assign nodes in Left set to Right set of nodes. There is a "b" constraint, which limits the maximum possible node degrees on the Right set. Since it is a ...
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1answer
25 views

Best and Worst options in Gale Shapley algorithm for an agent

Please consider the figure below. I have to find the best and worst options for W. From the preference list of W, the best option for W is D but there is no matching of W with D in all the 6 options. ...
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1answer
68 views

An algorithm to find the closest match between 2 arrays of RGB pixel tuples

So I'm looking for a bit of an abstract algorithm and I'd appreciate any references to read up on. This is a bit tough to explain but I'll try my best. Suppose we have 2 arrays of ...
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1answer
19 views

Problem with understanding two sided Matching Algorithm: maximium cardinality

I am trying to understand the maximum cardinality problem in the context of stable matching algorithm. I am reading the following article at the link: A Two-Sided Matching Decision Model Based on ...
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Blossom's Algorithm or Maximal Matching [closed]

I am given a complete weighted graph and I need to make pairs among the vertices so that the sum of weights is maximum. (If I have vertices $v1, v2, v3, v4$ and if I make pairs $(v1,v2)$ and $(v3,v4)$ ...
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Relation between shape descriptors and featured connected component in matching problems

Hope I'm asking in the correct community, from the title, I need a clarification regarding the idea of dealing with 3d descriptor and featured connected components. I have this approach: model -> ...
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1answer
385 views

one-to-many matching in bipartite graphs?

Consider having two sets $L$ (left) and $R$ (right). $R$ nodes have a capacity limit. Each edge $e$ has a cost $w(e)$. I want to map each of the $L$ vertices to one node from $R$ (one-to-many ...
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1answer
293 views

How to find maximum matching edges in undirected tree

Let $B$ be an undirected tree with $|V|$ nodes given as adjacency list. I want to develop a greedy algorithm using pseudo code to find a maximal matching in runtime $\mathcal{O}(|V|)$. My approach: ...
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1answer
31 views

Computing minimum partition of poset of $N$ intervals into chains in $o(N^{2.5})$ time?

Consider a set $P$ of $N$ intervals $\{I_i = (l_i, r_i)\}$ partially ordered according the standard interval order: $I_i < I_j$ iff $r_i \le l_j$. I want to find a minimum cardinality partition of ...
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1answer
824 views

A problem with the greedy approach to finding a maximal matching

Suppose I have an undirected graph with four vertices $a,b,c,d$ which are connected as in a simple path from $a$ to $d$, i.e. the edge set $\{(a,b), (b,c), (c,d)\}$. Then I have seen the following ...
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String matching problem needed some explanation

This is a question from CLRS book. (Chapter 32, string matching, the question is the problem for the whole chapter, it's in the end of the chapter) Let $y^i$ denote the concatenation of string y ...
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15 views

Maximum cardinality bipartite matching when nodes are ordered and only subsets can be matched?

Maximum bipartite matchine problem can be converted to the maximum flow problem and it can be solved by Edmonds-Karp algorithm in O(VE)<=O(V^3). But there can be bounded problem, when the nodes on ...
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1answer
56 views

Matching each unique pokemon with a unique move

Background: There are 832 unique Pokemon in the Pokemon universe, and There are 728 fighting moves that the Pokemon can collectively learn. No Pokemon can learn to do every fighting move, and some ...
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1answer
58 views

Give an example of a connected graph where α(G) =100 and β(G) = 200

So I need to find a form of a graph such that its vertex cover is twice that of its matching, but I am running into problems brainstorming, I know K3 is of this form, but not one at such a magnitude.
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60 views

Matching of two weighted graphs allowing one-to-many mapping

I am looking for a heuristic for a graph matching problem as follows. Given two graphs: $A$ (consisting of nodes $a_i$) and $B$ (consisting of nodes $b_i$). Typically the size of $B$ is larger than ...
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1answer
38 views

Prove that any tree contains a matching of size |InternalNodes|/2 [closed]

How can i prove that any tree contains a matching of size |InternalNodes|/2? Thanks in advance
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1answer
86 views

Term for a graph decomposition based on a maximum matching

Let $M$ be a maximum cardinality matching in a bipartite graph $G(X+Y,E)$. Let $X_0$ be the subset of $X$ unmatched by $M$. Define the following sequence: $Y_1 = $ the neighbors of $X_0$ using edges ...
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148 views

Maximum weighted matching for directed (non-bipartite) graphs

This post concerns mainly non-bipartite graphs. Edmonds (1961) have proposed the Blossom algorithm to solve the maximum matching problem for undirected graphs. The best implementation of it is due ...
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1answer
224 views

Matching Algorithm - How to maximize matched quantity with unique matching rules?

Given a set $S=\{A,B,\cdots,H\}$. Elements in $S$ can be matched according to the following rules: $$\begin{aligned} A\leftrightarrow B\\ C\leftrightarrow D\\ B+C\leftrightarrow F\\ D+A\...
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1answer
40 views

How to construct an ordinary matching from a fractional matching?

Given a graph $G=(V,E)$. A fractional matching, say $f$, assigns every edge $e \in E$ to a fraction $f(e) \in [0,1]$, with the constraint: for $v \in V$, $\sum_{e \ni v}f(e) \leq 1 $. My question ...
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37 views

Online Many-to-one Matching

Offline Problem I have a graph $\mathcal{G} = (\mathcal{D} \cup \mathcal{A}, \mathcal{E})$. Each edge $e \in \mathcal{E}$ between the two vertex sets $\mathcal{D}$ and $\mathcal{A}$ has an ...
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0answers
273 views

Perfect matching in complete, weighted graph

I'm trying to find pairs in a complete, weighted graph, similar to the one below (weights not shown). For each possible pair there is a weight and I would like to find pairs for including all ...
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0answers
23 views

Optimal matching of individuals in vehicles

I am looking for an algorithm to find the optimal matching/allocation of n individuals in m identical vehicles. The aim is to create groups of individuals who will share these vehicles. Groups' size ...
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29 views

Bipartite vertex cover [duplicate]

If this link can be any help https://codeforces.com/blog/entry/63164 A vertex cover of a graph is a set of vertices such that each edge of the graph is incident to at least one vertex of the set. A ...
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112 views

Find a minimum weight matching of a specific size

Given a positive-weighted complete graph $G=(V,E)$ and an even integer $k$, I want to find a minimum weight matching of size exactly $k$, i.e., I want to choose $k/2$ vertex disjoint edges such that ...
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3answers
1k views

What is a fractional matching?

For the maximum matching problem, we can find the fractional matching which I understand involves some sort of weighting for the edges. However, I cannot seem to find an exact and simple explanation ...
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134 views

Maximum matching blossom algorithm

https://stanford.edu/~rezab/classes/cme323/S16/projects_reports/shoemaker_vare.pdf Why do we use blossom contrast in maximum matching? The examples that I have seen can easily be solved by just ...
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2answers
45 views

Minimizing catastrophic risk in Gale-Shapley matching

In the hospital-resident assignment problem we have to match a large set of med students with a small set of hospitals. Hospitals may accept multiple students, but the number of students is much ...
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0answers
40 views

Find maximal matching in tree in $O\left(\frac{n}{\log n}\right)$

As any tree can be described as a binary sequence ($i$-th bit is 0 if the edge goes down and 1 otherwise, every edge is travelled twice $-$ up and down, so such sequence's length is $2 |V| - 2$), any ...
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Computing a shortest $M$-alternating walk (for the Blossom algorithm)

When explaining the Blossom algorithm for maximum (nonbipartite) matchings, Shrijver describes, given a simple graph $G = (V,E)$, a matching $M \subseteq E$, and the set $X \subseteq V$ of nodes ...
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1answer
60 views

Can maximum matching algorithms be used for maximum weight matching?

There are two fast algorithms for maximum matching on general graphs: Micali and Vazirani in $O(E\sqrt{V})$. Mucha and Sankowski in $O(V^{2.376})$. Can these be also used for maximum weighted ...
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1answer
177 views

Find a minimum-weight perfect b-matching, where b is even

How would one find a minimum-weight perfect b-matching of a general graph, where the number of edges incident on each vertex is a positive even number not greater than b? A minimum-weight perfect b-...
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1answer
155 views

Find a minimum-weight perfect 2-matching

How would one find a minimum-weight perfect 2-matching of a general graph? Is it possible to use standard matching techniques like Blossom V? A minimum-weight perfect 2-matching of a graph G is a ...